Complex dynamical behaviors in a Bertrand game with service factor and differentiated products

In this paper, taking the factor of product service provided by manufacturers into consideration, a static Bertrand duopoly game with is studied first, in which these two oligarchs produce differentiated products. A dynamic bounded rationality established using gradient mechanism. Keeping adjustment speeds relatively small range may help long-term stable operation market. It found that there another 2-cycle, different from one flip bifurcated fixed point, appear through saddle-node bifurcation. The unstable set saddle cycle, connecting to node, gives closed invariant curve. addition, emergence intermittent chaos implies economic system has capability self-regulating, where PM-I intermittency and crisis-induced have been studied. With critical curves, qualitative changes on basin attraction are investigated.